login
A190275
Semiprimes of the form p*(p^2 - p + 1).
6
6, 21, 301, 2041, 296341, 486877, 2666437, 3420301, 4304341, 7152001, 38159521, 42387097, 54296677, 95235601, 158048281, 229971241, 265434901, 383712781, 454166017, 775307917, 972261181, 1063290841, 1304557801, 1392422041, 1730882401, 1863895261, 2631883561, 2879450461, 3714274297, 3845297341, 4070454361, 4256780041, 4849695001, 5328809461, 5722533337, 5838483601, 7218898681, 7841065621
OFFSET
1,1
COMMENTS
This sequence is infinite, assuming Schinzel's Hypothesis H.
Related to Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q + r + 1") setting p = q. Generalization can be achieved by removing semiprimality condition and accepting p^e, e >= 2.
These are semiprimes m = p*q such that 1/p + 1/q - 1/m = p/q. Cf. A326690. - Amiram Eldar and Thomas Ordowski, Jul 22 2019
LINKS
For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.
EXAMPLE
a(1) = 6 = 2*3 = 2*(2^2-2+1).
a(2) = 21 = 3*7 = 3*(3^2-3+1).
a(3) = 301 = 7*43 = 7*(7^2-7+1).
MAPLE
seq(`if`(isprime((ithprime(i)^2-ithprime(i)+1))=true, (ithprime(i)^2-ithprime(i)+1)*ithprime(i), NULL), i=1..300);
MATHEMATICA
p = Select[Prime@ Range@ 500, PrimeQ[#^2 - # + 1] &]; p (p^2 - p + 1) (* Giovanni Resta, Jul 22 2019 *)
PROG
(PARI) forprime(p=2, 1e4, if(isprime(k=p^2-p+1), print1(p*k", "))) \\ Charles R Greathouse IV, May 08 2011
CROSSREFS
Cf. A065508 (primes p such that p^2-p+1 is prime).
Cf. A001358 (semiprime), A003415 (arithmetic derivative), A164643, A190272 (n'=a-1), A190273 (n'=a+1), A190274 (n'=p^2-1).
Sequence in context: A143049 A213680 A164643 * A261844 A372425 A007594
KEYWORD
nonn
AUTHOR
Giorgio Balzarotti, May 07 2011
STATUS
approved